Third seed Medvedev was on the brink of victory on Thursday evening before Mannarino staged a late fightback to level the third set at 4-4 and take the match into a second day.
The world number 35 continued to keep pace with the Russian, who has never advanced to the quarter-finals at the All England Club.
Medvedev clinched the tiebreak when the left-handed Mannarino sent a forehand long and the Russian will next play American Marcos Giron or Hungarian Marton Fucsovics.
Medvedev, who has not dropped a set at the tournament, said he may have a new lucky ritual to go through if he has to play a match over two days again.
"I remember a couple of years ago I had this against (Hubert) Hurkacz and I lost and I kind of prepared in a way for a new match. I did an ice bath, I finished the match as if it was a normal match and the next day I had to play again," he said in an on-court interview.
"And this didn't work out for me, so I changed it. I didn't do anything as if the match was finished, I didn't wash my hair, I was like 'I'll wash it after the match' and it worked so I'm going to continue like this if next time I have a two-day match.
"I did take a shower," he added.
